Optimality Conditions For Interval-valued Programming

Uncertain programming is an important branch in Operations Research,whose objective functions or constrain functions have uncertain parameters and widely exists in practical optimization problems.As a special uncertainty optimization,interval-valued programming can effectively deal with optimization problems with uncertain parameters.The optimality condition plays an important role in the study of the existence of interval-valued programming solutions.This dissertation mainly studies the KKT condition,AKKT condition and CAKKT condition with interval-valued programming.It is organized as follows:In Chapter 1,we introduce the research background of interval-valued programming,the research situation of optimality conditions for interval-valued programming,and the research situation of AKKT conditions.In Chapter 2,we review the theory of interval analysis and nonsmooth analysis,and list some common symbols which used in this dissertation.In Chapter 3,under the assumption that the objective interval-valued function is weakly continuously differentiable,we study the KKT and CAKKT optimality conditions of the single-objective interval-valued programming.First,it was shown that the KKT condition is a necessary condition for a local LU-solution of the single-objective interval-valued programming under Mangasarian-Fromovitz constraint qualification(MFCQ for short).Second,we define the CAKKT condition for the single-objective interval-valued programming.Then we prove that the CAKKT condition does not require any constraint qualifications to become a necessary condition for a local LU-solution.Finally,it was shown that the CAKKT condition is a sufficient condition for an LU-solution under the convexity assumption.The results presented in this dissertation generalize known results from scalar optimization problem to interval-valued optimization problem.In Chapter 4,under the assumption that the objective interval-valued function is a nonsmooth function,we study the Fritz John,KKT and AKKT conditions of the multiobjective interval-valued optimization problem.First,we show that a local weak LU-efficient solution must satisfy the Fritz John and KKT conditions through the generalized Jacobian matrix of the nondifferentiable vector-valued function.Secondly,we introduce the definition of the AKKT condition for the multiobjective interval-valued programming.Then,we prove that the AKKT condition becomes a necessary condition for a local weak LU-efficient solution of multiobjective interval-valued programming without any constraint specification.Finally,we prove that the AKKT condition is a sufficient condition for a weak LU-efficient solution under the convexity assumption.This Chapter is an extension of the content of Chapter 3,and the model is more widely applicable.In Chapter 5,we summarize the main work and the research plan in next step.